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For a positive integer n show that (1+i)^n+(1-i)^n=2^((n+2)/2) "cos((npi)/4) |
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Answer» Solution :`L.H.S.=(1+i)^n+(1-i)^n` `={sqrt2(1/sqrt2^1+i1/sqrt2)}^n+{sqrt2(1/sqrt2-i1/sqrt2)}^n` `=2^(n//2)[("cos"pi/4="isin"pi/4)^n="cos"pi/4 - "isin"pi/4)^n]` `=2^(n//2)xx2"cos"(NPI)/4=2^(n//2+1)"cos"(npi)/4` `=2^((n+2)/2) "cos"(npi)/4=R.H.S` |
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